This is a post detailing the design of a low delta-V mission from Earth to Mercury.
As many Orbiter users will know, Mercury is a challenging destination because of its orbital location deep inside the Sun's gravitational well. A direct Hohmann transfer from Earth to Venus requires an ejection burn of around 5.5 km/s; and around 7 km/s orbit injection burn at Mercury to enter into a low orbit around the planet. In total, roughly 12.5 km/s of delta-V is required. This is a prodigious amount of fuel. Surely, there must be a better,cheaper way of getting to Mercury?
Indeed there is - the MESSENGER mission (NASA) to Mercury is evidence of that. From Wikipedia: "MESSENGER extensively used gravity assist manoeuvres at Earth, Venus, and Mercury to reduce the speed relative to Mercury, then used its large rocket engine to enter into an elliptical orbit around the planet. The multi-flyby process greatly reduced the amount of propellant necessary to slow the spacecraft, but at the cost of prolonging the trip by many years and to a total distance of 4.9 billion miles. To further minimise the amount of necessary propellant, the spacecraft orbital insertion targeted a highly elliptical orbit around Mercury.".
But to understand what this 'better way' of getting to Mercury (and why it works), it helps to resort (once again) to the Tisserand Plot. With the Tisserand Plot, we can build a MESSENGER-like mission to Mercury. The resulting mission design will look like the following:
To recap early posts: the Tisserand Plot is a succinct way of representing orbits (in this case, about the Sun) on a two-dimensional graph. It plots orbital periapsis and apoapsis on a two-dimensional graph. On it, any orbit around the Sun can be represented as a point.
Tisserand Plots were introduced introduced in http://www.orbiter-forum.com/showthread.php?t=36461; and used to design a simple Earth Gravity Assist (EGA) mission in http://www.orbiter-forum.com/showthread.php?t=36562.
In this post, the Tisserand Plot again forms the basis for the design of a more complicated, low-cost mission to Mercury. To make life simple for ourselves, we are going to make the bold (and somewhat unrealistic) assumption that the Earth, Venus and Mercury all move around the Sun in circular orbits with the following parameters:
Mean orbital radius:
Earth 1.00 AU
Venus 0.72 AU
Mercury 0.39 AU
Mean orbital speed:
Earth 29.78 km/s
Venus 35.02 km/s
Mercury 47.36 km/s
The Standard Hohmann transfer from Earth to Mercury
To start our analysis, it may be useful to see how the standard Hohmann transfer from Earth to Venus is represented on a Tisserand Plot (below)
On this plot, there are two lines of constant hyperbolic excess velocity ([MATH]v_\infty[/MATH]). The red line is a line of constant hyperbolic excess velocity for an encounter with Earth; and the green line is a line of constant hyperbolic excess velocity for an encounter with Venus. These two lines are the lines with the lowest hyperbolic excess velocity that we can construct that still (just) intersect. The point of intersection, the blue dot, is the Hohmann transfer orbit (remember: orbits are represented as 'points' on the Tisserand Plot). It sits at the intersection of the [MATH]v_\infty = 7.47\,km/s[/MATH] curve for Earth and the [MATH]v_\infty=9.42 km/s[/MATH] curve for Mercury. This means that for a Hohmann transfer orbit, once we manage to escape the Earth's gravitational well, we need to have a hyperbolic excess velocity (relative to Earth) of 7.57 km/s. And on our approach to Mercury and before falling into its gravity well, we will find our speed will be 9.42 km/s. To leave Earth with a hyperbolic excess velocity of 7.57 km/s, we need an Earth escape burn of around 5.5 km/s; and once we arrive at Mercury, to achieve a low circular orbit around that planet, we need an orbit insertion burn of around 7 km/s.
Of course, this analysis assumes circular orbits and coplanar movement of Earth and Jupiter so, in practice, we can do a bit better than this if we arrive at Mercury at its periapsis, the above analysis gets the numbers broadly right.
A better strategy for getting to Mercury
A better strategy for getting to Mercury is to take advantage of the fact that Venus orbits between Earth and Mercury. One of the reasons for the inefficiency of the Earth/Mercury Hohmann transfer is that we have to lower the periapsis of our orbit all the way down to Mercury's orbital radius. This takes a lot of delta-V to slower the spacecraft in its orbit around the Sun to the point where perihelion intersects with the orbit of Mercury. We can save a lot of fuel if we lower the periapsis of our orbit down only to Venus and then use ballistic gravity assists around Venus to further lower our orbital trajectory all the way down to Mercury's orbit.
Assuming we get our timing right and encounter Mercury close to the periapsis of our orbit, then we can use another series of Mercury flybys coupled with low cost Deep Space Manoeuvres (DSM) to dump kinetic energy and further reduce our Mercury encounter velocity.
This, in a nutshell, is what we are hoping to achieve with the mission design. How does this translate into the graphical representation of the Tissrand Plot?
Building the Tisserand Plot for the improved Earth to Mercury transfer strategy
The cornerstone of building the improved transfer strategy is the point 'C' on the first graph above. This point represents a Hohmann transfer trajectory from Venus to Mercury. On this transfer orbit, upon departure from Venus, the hyperbolic excess velocity is 5.60 km/s; and on arrival at Mercury, the hyperbolic excess velocity is 6.59 km/s (assuming a circular orbit). This accounts, then, for the blue line on the plot as well as the lowest of the four green lines.
How about the transfer orbit from Earth to Venus? To obtain this transfer orbit, we identify the line of constant hyperbolic excess velocity (relative to Earth) such that it terminates on the left-hand side of the graph just at the blue line (the line of constant hyperbolic velocity relative to Venus corresponding to the Hohmann transfer orbit from Venus to Mercury). By playing around with test values of [MATH]v_\infty[/MATH], we quickly find that the value of the hyperbolic excess velocity (relative to Earth) that we need is 3.10 km/s. On the Tisserand Plot, the Earth to Venus transfer orbit is given by the point 'A' which has an orbital periapsis of around 0.67 AU.
So, how do we get from orbit 'A' to orbit 'C'? The movement from the point 'A' to 'C' represents one or more ballistic gravitational encounters of our spacecraft with Venus. If we can get the timing to work, and if the encounter doesn't require that we dive into the Venusian atmosphere, then we can transfer from orbit 'A' to orbit 'C' with just one flyby. If not, then we can transform our orbit to orbit 'C' with two or more ballistic flybys. For example, on the first flyby, we could transform our orbit to orbit 'B' and then one Venusian year later, we could arrange for a second Venus flyby that transforms our orbit from orbit 'B' to orbit 'C'.
So far, we understand the rationale for the red, the blue and lowest of the four green lines. But what about the three other green lines on the Tisserand Plot? In our mission design, we have our spacecraft encountering Mercury with a hyperbolic excess velocity of 6.59 km/s. Although this is considerably less than the 9.42 km/s of the direct Hohmann transfer, it is still a high approach velocity. We can do better than this by including a few v-infinity leveraged transfers (VILTs) at the backend of the mission. These VILTs are designed to efficiently dump hyperbolic excess velocity by performing a small corrective burn at the aphelion of each orbit. So, the sequence that we wish to implement these VILTs:
1. A ballistic flyby of Mercury along the [MATH]v_\infty=6.59\,km/s[/MATH] line to orbit 'D' long roughly on the line of 3:2 resonance with Mercury.
2. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'E'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=5.40\,km/s[/MATH].
3. At the next encounter with Mercury, use the ballistic flyby to move us along the [MATH]v_\infty=5.40\,km/s[/MATH] line from orbit 'E' to 'orbit 'F' which lies on the line of 4:3 resonance with Mercury.
4. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'G'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=4.08\,km/s[/MATH].
5. At the next encounter with Mercury, use the ballistic flyby to move us along the [MATH]v_\infty=4.08\,km/s[/MATH] line from orbit 'G' to 'orbit 'H' which lies on the line of 6:5 resonance with Mercury.
6. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'H'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=2.75\,km/s[/MATH] and orbit 'I'.
7. Now, finally when the spacecraft next encounters Mercury, it executes a retrograde burn of around 2.1 km/s to enter into low orbit around Mercury.
Next steps
The above mission design sets out a blue-print for a low delta-v mission to Mercury. Although we haven't copied the MESSENGER mission, it turns out that the underling logic of that mission is much the same - and, therefore, achieves similar results.
At this stage, though, to implement this in Orbiter one has to get down to the hard graft of working out the exact timings for each stage of this mission and ensuring that all of its component pieces 'work'. This, however, is a subject for a subsequent post.
As many Orbiter users will know, Mercury is a challenging destination because of its orbital location deep inside the Sun's gravitational well. A direct Hohmann transfer from Earth to Venus requires an ejection burn of around 5.5 km/s; and around 7 km/s orbit injection burn at Mercury to enter into a low orbit around the planet. In total, roughly 12.5 km/s of delta-V is required. This is a prodigious amount of fuel. Surely, there must be a better,cheaper way of getting to Mercury?
Indeed there is - the MESSENGER mission (NASA) to Mercury is evidence of that. From Wikipedia: "MESSENGER extensively used gravity assist manoeuvres at Earth, Venus, and Mercury to reduce the speed relative to Mercury, then used its large rocket engine to enter into an elliptical orbit around the planet. The multi-flyby process greatly reduced the amount of propellant necessary to slow the spacecraft, but at the cost of prolonging the trip by many years and to a total distance of 4.9 billion miles. To further minimise the amount of necessary propellant, the spacecraft orbital insertion targeted a highly elliptical orbit around Mercury.".
But to understand what this 'better way' of getting to Mercury (and why it works), it helps to resort (once again) to the Tisserand Plot. With the Tisserand Plot, we can build a MESSENGER-like mission to Mercury. The resulting mission design will look like the following:
To recap early posts: the Tisserand Plot is a succinct way of representing orbits (in this case, about the Sun) on a two-dimensional graph. It plots orbital periapsis and apoapsis on a two-dimensional graph. On it, any orbit around the Sun can be represented as a point.
Tisserand Plots were introduced introduced in http://www.orbiter-forum.com/showthread.php?t=36461; and used to design a simple Earth Gravity Assist (EGA) mission in http://www.orbiter-forum.com/showthread.php?t=36562.
In this post, the Tisserand Plot again forms the basis for the design of a more complicated, low-cost mission to Mercury. To make life simple for ourselves, we are going to make the bold (and somewhat unrealistic) assumption that the Earth, Venus and Mercury all move around the Sun in circular orbits with the following parameters:
Mean orbital radius:
Earth 1.00 AU
Venus 0.72 AU
Mercury 0.39 AU
Mean orbital speed:
Earth 29.78 km/s
Venus 35.02 km/s
Mercury 47.36 km/s
The Standard Hohmann transfer from Earth to Mercury
To start our analysis, it may be useful to see how the standard Hohmann transfer from Earth to Venus is represented on a Tisserand Plot (below)
On this plot, there are two lines of constant hyperbolic excess velocity ([MATH]v_\infty[/MATH]). The red line is a line of constant hyperbolic excess velocity for an encounter with Earth; and the green line is a line of constant hyperbolic excess velocity for an encounter with Venus. These two lines are the lines with the lowest hyperbolic excess velocity that we can construct that still (just) intersect. The point of intersection, the blue dot, is the Hohmann transfer orbit (remember: orbits are represented as 'points' on the Tisserand Plot). It sits at the intersection of the [MATH]v_\infty = 7.47\,km/s[/MATH] curve for Earth and the [MATH]v_\infty=9.42 km/s[/MATH] curve for Mercury. This means that for a Hohmann transfer orbit, once we manage to escape the Earth's gravitational well, we need to have a hyperbolic excess velocity (relative to Earth) of 7.57 km/s. And on our approach to Mercury and before falling into its gravity well, we will find our speed will be 9.42 km/s. To leave Earth with a hyperbolic excess velocity of 7.57 km/s, we need an Earth escape burn of around 5.5 km/s; and once we arrive at Mercury, to achieve a low circular orbit around that planet, we need an orbit insertion burn of around 7 km/s.
Of course, this analysis assumes circular orbits and coplanar movement of Earth and Jupiter so, in practice, we can do a bit better than this if we arrive at Mercury at its periapsis, the above analysis gets the numbers broadly right.
A better strategy for getting to Mercury
A better strategy for getting to Mercury is to take advantage of the fact that Venus orbits between Earth and Mercury. One of the reasons for the inefficiency of the Earth/Mercury Hohmann transfer is that we have to lower the periapsis of our orbit all the way down to Mercury's orbital radius. This takes a lot of delta-V to slower the spacecraft in its orbit around the Sun to the point where perihelion intersects with the orbit of Mercury. We can save a lot of fuel if we lower the periapsis of our orbit down only to Venus and then use ballistic gravity assists around Venus to further lower our orbital trajectory all the way down to Mercury's orbit.
Assuming we get our timing right and encounter Mercury close to the periapsis of our orbit, then we can use another series of Mercury flybys coupled with low cost Deep Space Manoeuvres (DSM) to dump kinetic energy and further reduce our Mercury encounter velocity.
This, in a nutshell, is what we are hoping to achieve with the mission design. How does this translate into the graphical representation of the Tissrand Plot?
Building the Tisserand Plot for the improved Earth to Mercury transfer strategy
The cornerstone of building the improved transfer strategy is the point 'C' on the first graph above. This point represents a Hohmann transfer trajectory from Venus to Mercury. On this transfer orbit, upon departure from Venus, the hyperbolic excess velocity is 5.60 km/s; and on arrival at Mercury, the hyperbolic excess velocity is 6.59 km/s (assuming a circular orbit). This accounts, then, for the blue line on the plot as well as the lowest of the four green lines.
How about the transfer orbit from Earth to Venus? To obtain this transfer orbit, we identify the line of constant hyperbolic excess velocity (relative to Earth) such that it terminates on the left-hand side of the graph just at the blue line (the line of constant hyperbolic velocity relative to Venus corresponding to the Hohmann transfer orbit from Venus to Mercury). By playing around with test values of [MATH]v_\infty[/MATH], we quickly find that the value of the hyperbolic excess velocity (relative to Earth) that we need is 3.10 km/s. On the Tisserand Plot, the Earth to Venus transfer orbit is given by the point 'A' which has an orbital periapsis of around 0.67 AU.
So, how do we get from orbit 'A' to orbit 'C'? The movement from the point 'A' to 'C' represents one or more ballistic gravitational encounters of our spacecraft with Venus. If we can get the timing to work, and if the encounter doesn't require that we dive into the Venusian atmosphere, then we can transfer from orbit 'A' to orbit 'C' with just one flyby. If not, then we can transform our orbit to orbit 'C' with two or more ballistic flybys. For example, on the first flyby, we could transform our orbit to orbit 'B' and then one Venusian year later, we could arrange for a second Venus flyby that transforms our orbit from orbit 'B' to orbit 'C'.
So far, we understand the rationale for the red, the blue and lowest of the four green lines. But what about the three other green lines on the Tisserand Plot? In our mission design, we have our spacecraft encountering Mercury with a hyperbolic excess velocity of 6.59 km/s. Although this is considerably less than the 9.42 km/s of the direct Hohmann transfer, it is still a high approach velocity. We can do better than this by including a few v-infinity leveraged transfers (VILTs) at the backend of the mission. These VILTs are designed to efficiently dump hyperbolic excess velocity by performing a small corrective burn at the aphelion of each orbit. So, the sequence that we wish to implement these VILTs:
1. A ballistic flyby of Mercury along the [MATH]v_\infty=6.59\,km/s[/MATH] line to orbit 'D' long roughly on the line of 3:2 resonance with Mercury.
2. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'E'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=5.40\,km/s[/MATH].
3. At the next encounter with Mercury, use the ballistic flyby to move us along the [MATH]v_\infty=5.40\,km/s[/MATH] line from orbit 'E' to 'orbit 'F' which lies on the line of 4:3 resonance with Mercury.
4. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'G'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=4.08\,km/s[/MATH].
5. At the next encounter with Mercury, use the ballistic flyby to move us along the [MATH]v_\infty=4.08\,km/s[/MATH] line from orbit 'G' to 'orbit 'H' which lies on the line of 6:5 resonance with Mercury.
6. At the aphelion of the new orbit, execute a small prograde burn to raise orbital periapsis back up to Mercury's orbital radius - i.e., orbit 'H'. This shifts the orbit to the line of constant hyperbolic excess velocity with [MATH]v_\infty=2.75\,km/s[/MATH] and orbit 'I'.
7. Now, finally when the spacecraft next encounters Mercury, it executes a retrograde burn of around 2.1 km/s to enter into low orbit around Mercury.
Next steps
The above mission design sets out a blue-print for a low delta-v mission to Mercury. Although we haven't copied the MESSENGER mission, it turns out that the underling logic of that mission is much the same - and, therefore, achieves similar results.
At this stage, though, to implement this in Orbiter one has to get down to the hard graft of working out the exact timings for each stage of this mission and ensuring that all of its component pieces 'work'. This, however, is a subject for a subsequent post.
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